LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Base change for intertwiner spaces

Statement

Let $E / F$ be a field extension, $G$ a group, and $V, W$ finite-dimensional $F$ -representations of $G$ . The map $E \otimes_{F} {Hom}_{G} (V, W) ⟶ {Hom}_{G} (E \otimes_{F} V, E \otimes_{F} W), a \otimes f ⟼ ((b \otimes v) \mapsto a b \otimes f (v))$ is an $E$ -linear isomorphism.

Facts & Assumptions

Given: $E / F$ , $G$ , $V$ , and $W$ as in the statement.

[L1]

Extension of scalars sends an $F$ -linear map $f$ to $1_{E} \otimes f$ (Restriction of scalars and extension of scalars $S \otimes_{R} M$ along a ring homomorphism $R \to S$ ).

[L2]

An intertwiner is exactly a linear map satisfying $f ρ_{V} (g) = ρ_{W} (g) f$ for every $g \in G$ (Intertwiners, the spaces ${Hom}_{G} (V, W)$ and ${End}_{G} (V)$ , equivalent representations, and faithful representations).

Proof

technique · direct

1.1

Choose $F$ -bases of $V$ and $W$ . By [L2], ${Hom}_{G} (V, W)$ is the simultaneous kernel in ${Hom}_{F} (V, W)$ of the maps $f \mapsto f ρ_{V} (g) - ρ_{W} (g) f$ .

L2choose

2.1

Tensoring a kernel of maps between finite-dimensional $F$ -spaces with the field $E$ preserves that kernel, because tensoring with a field extension is exact. The simultaneous kernel after tensoring is, by [L2], precisely ${Hom}_{G} (E \otimes_{F} V, E \otimes_{F} W)$ .

L2step 1.1

3.1

The resulting identification sends $a \otimes f$ to the displayed map, which agrees with [L1]. Therefore it is the asserted $E$ -linear isomorphism.

L1step 2.1∎

Depends on

Used by

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gabor Wiese, Galois Representations, Theorem 2.2.4 (standard reference, not scraped)
Weizhe Zheng, Lectures on Algebra, proof of Proposition 4.2.8 (standard reference, not scraped)